Find the indicated probability: P(Z>2.24) (a) .9875 (b) .0119 (c).0125 (d) .1075

Answer 1

Since the normal distribution is symmetric P(Z > 2.24) = P(Z < -2.24) = 0.0125 (c)

Answer 2

All of the questions you posted use the exact same ideology, so I'll answer one, and you can message me if you need help with the others. Hopefully this helps. :)

All a Z-score is is a certain "cutoff point" on a standard normal distribution -- this is a normal distribution with mean 0 and standard deviation 1. This means that the distribution itself represents the probability of getting something (since the area under the entire curve is 1). So, since we know that, for a normal distribution, area = probability, what this question is asking is -- what is probability that we get a Z-score greater than 2.24? Aka: in a standard normal distribution, what is the area from 2.24 to infinity?

To solve this, there are two options.

Solution 1

The first way, and probably the way they want you to solve this, is using a Z-score table. Here is a good one: http://pages.stat.wisc.edu/~ifischer/Statistical_Tables/Z-distribution.pdf

First, we find the Z-score of interest. In this case, it is 2.24. We find that the probability that Z ≤ 2.24 = 0.98745. Since we are concerned with what is greater than 2.24, we subtract this value from 1. Thus:

P(Z > 2.24) =

1 - P(Z≤2.24) =

1 - 0.98745 =

0.01255

This leads us to answer c). Alternatively, with this method, since we know a normal distribution is symmetric, the P(Z≤-2.24) = P(Z>2.24), which we can find using the table as well. This gives us the same answer if we look at our table: .01255.

Solution 2

The second solution would be to use a command on your calculator, such as a TI-84. You could also use an online calculator like this one: http://onlinestatbook.com/2/calculators/normal_dist.html

We know the mean = 0, standard deviation = 1, since we are dealing with a standard normal distribution (that is what a Z score uses). We are looking for above 2.24, and this gives us our answer: .0125.